Preheat the oven to 400 degrees F. Line 2 baking sheets with parchment paper.
In a small bowl, toss together the cheeses, herbs and cracked black pepper. Reserve 2 tablespoons of the cheese mixture.
In a separate small bowl, whisk the egg with 2 teaspoons water to make an egg wash.
Roll the puff pastry out on a floured surface to a rectangle about 12-by-10 1/2 inches. Brush the entire surface with egg wash. Cut the pastry in half crosswise forming 2 rectangles. Sprinkle the herb cheese mixture over one rectangle and top with the other. Using the rolling pin, roll the pastry just to help the 2 rectangles adhere. Using a pizza wheel or a sharp knife, cut the layered pastry into 1/2-inch-wide strips; you'll get about 24. Twist each strip and transfer it to a prepared baking sheet, pinching each of the ends tightly, spacing the strips about 2 inches apart from one another. Brush strips with egg wash and sprinkle with reserved cheese mixture.
Bake until golden brown, 10 to 12 minutes. Cool completely on a wire rack before serving.
The assembled cheese straws can be frozen before baking for up to a month. When baking, they can go straight from the freezer to the oven.